Vertex operator algebra

Definition

A vertex operator algebra over $\C$ consists of:

• a $\Z$-graded complex vector space $V = \bigoplus_{n \in \Z} V_n$ with $\dim V_n < \infty$ for all $n$ and $V_n = 0$ for all sufficiently negative $n$;
• a linear map $Y \colon V \to (\End V)[[z, z^{-1}]\,]$, written $Y(v, z) = \sum_{n \in \Z} v_n z^{-n-1}$ with $v_n \in \End V$, into formal series in a variable $z$;
• a vector $\mathbf{1} \in V_0$, the vacuum vector, and a vector $\omega \in V_2$, the conformal vector.

Write $\delta(x) = \sum_{n \in \Z} x^n$, and expand each binomial $(x_1 - x_2)^m$ in nonnegative powers of $x_2$. These data must satisfy:

1. for all $u, v \in V$, $u_n v = 0$ for all sufficiently large $n$;
2. $Y(\mathbf{1}, z) = \id_V$, and for all $v \in V$, $Y(v, z)\mathbf{1}$ has only nonnegative powers of $z$ and its constant term is $v$;
3. (Jacobi identity) for all $u, v \in V$, in formal variables $z_0, z_1, z_2$, $$z_0^{-1} \delta\!\left(\frac{z_1 - z_2}{z_0}\right) Y(u, z_1) Y(v, z_2) - z_0^{-1} \delta\!\left(\frac{z_2 - z_1}{-z_0}\right) Y(v, z_2) Y(u, z_1) = z_2^{-1} \delta\!\left(\frac{z_1 - z_0}{z_2}\right) Y(Y(u, z_0) v, z_2);$$
4. writing $Y(\omega, z) = \sum_{n \in \Z} L(n) z^{-n-2}$, there is a number $c \in \C$, the central charge, with $[L(m), L(n)] = (m - n) L(m + n) + \frac{c}{12}(m^3 - m)\, \delta_{m+n, 0}\, \id_V$ for all $m, n \in \Z$;
5. $L(0) v = n v$ for all $n \in \Z$ and $v \in V_n$;
6. $Y(L(-1) v, z) = \frac{d}{dz} Y(v, z)$ for all $v \in V$.